Completely monotone function and Subordination: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Mateusz
(added "Examples" and "Properties")
 
imported>Mateusz
(Created page with "If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T...")
 
Line 1: Line 1:
A function $f : (0, \infty) \to [0, \infty)$ is said to be completely monotone (totally monotone, completely monotonic, totally monotonic) if $(-1)^k f^{(k)} \ge 0$ for $x > 0$ and $k = 0, 1, 2, ...$<ref name="SSV"/>
If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then
\[ \tilde{T}_t u = \int_{[0, \infty)} T_r u \mu_t(\mathrm d r) \]
defines another semigroup of operators $\tilde{T}_t$, which is said to be subordinate (in the sense of Bochner) to $\tilde{T}_t$.<ref name="S"/>


==Representation==
==Relation to Bernstein functions==
By Bernstein's theorem, a function $f$ is completely monotone if and only if it is the Laplace transform of a nonnegative measure,
For some [[Bernstein function]] $f$,
\[ f(z) = \int_{[0, \infty)} e^{-s z} m(\mathrm d s) . \]
\[ \int_{[0, \infty)} e^{-r z} \mu_t(\mathrm d r) = e^{-t f(z)} . \]
Here $m$ is an arbitrary Radon measure such that the above integral is finite for all $z > 0$.
Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.


==Examples==
One often writes $\tilde{L} = f(L)$, where $-L$ and $-\tilde{L}$ are the generators of $T_t$ and $\tilde{T}_t$, respectively. This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space.
The following functions of $z$ are completely monotone:
* $z^s$ for $s \le 0$,
* $e^{-t z}$ for $t \ge 0$,
* $\log(1 + \frac{1}{z})$,
* $e^{1 / z}$.
 
==Properties==
If $f_1, f_2$ are completely monotone and $c > 0$, then also $c f_1$, $f_1 + f_2$, $f_1 f_2$ are completely monotone.
 
If $f$ is completely monotone and $g$ is a [[Bernstein function]], then $f \circ g$ is completely monotone.


==References==
==References==
{{reflist|refs=
{{reflist|refs=
<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
<ref name="S">{{Citation | last1=Schilling | first1=R. | title=Subordination in the sense of Bochner and a related functional calculus | year=1998 | url=http://www.austms.org.au/Publ/Jamsa/V64P3/abs/p86/ | journal=J. Aust. Math. Soc. Ser. A | volume=64 | pages=368–396}}</ref>
}}
}}


{{stub}}
{{stub}}

Revision as of 06:07, 19 July 2012

If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T}_t u = \int_{[0, \infty)} T_r u \mu_t(\mathrm d r) \] defines another semigroup of operators $\tilde{T}_t$, which is said to be subordinate (in the sense of Bochner) to $\tilde{T}_t$.[1]

Relation to Bernstein functions

For some Bernstein function $f$, \[ \int_{[0, \infty)} e^{-r z} \mu_t(\mathrm d r) = e^{-t f(z)} . \] Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.

One often writes $\tilde{L} = f(L)$, where $-L$ and $-\tilde{L}$ are the generators of $T_t$ and $\tilde{T}_t$, respectively. This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space.

References

  1. ↑ Schilling, R. (1998), "Subordination in the sense of Bochner and a related functional calculus", J. Aust. Math. Soc. Ser. A 64: 368–396, http://www.austms.org.au/Publ/Jamsa/V64P3/abs/p86/ 

This article is a stub. You can help this nonlocal wiki by expanding it.

Retrieved from "https://web.ma.utexas.edu/mediawiki/index.php?title=Completely_monotone_function&oldid=1259"